Let's break down the expression step by step.

The given expression is:

\[
3.9 \times 0.5 + 4 \frac{5}{6} \div 3 \frac{3}{7}
\]

### Step 1: Multiplication
First, multiply \( 3.9 \times 0.5 \):

\[
3.9 \times 0.5 = 1.95
\]

### Step 2: Division of mixed numbers
Convert the mixed numbers into improper fractions:
- \( 4 \frac{5}{6} = \frac{29}{6} \)
- \( 3 \frac{3}{7} = \frac{24}{7} \)

Now, perform the division of the fractions:

\[
\frac{29}{6} \div \frac{24}{7} = \frac{29}{6} \times \frac{7}{24} = \frac{29 \times 7}{6 \times 24} = \frac{203}{144}
\]

### Step 3: Add the two results
Now, add \( 1.95 \) and \( \frac{203}{144} \):

Convert \( 1.95 \) to a fraction:

\[
1.95 = \frac{195}{100}
\]

We need a common denominator to add \( \frac{195}{100} \) and \( \frac{203}{144} \). The least common denominator of 100 and 144 is 720.

- Convert \( \frac{195}{100} \) to have a denominator of 720:

\[
\frac{195}{100} = \frac{195 \times 7.2}{100 \times 7.2} = \frac{1404}{720}
\]

- Convert \( \frac{203}{144} \) to have a denominator of 720:

\[
\frac{203}{144} = \frac{203 \times 5}{144 \times 5} = \frac{1015}{720}
\]

Now add:

\[
\frac{1404}{720} + \frac{1015}{720} = \frac{1404 + 1015}{720} = \frac{2419}{720}
\]

### Final Answer:
The result in fraction form is \( \frac{2419}{720} \), which is approximately:

\[
\frac{2419}{720} \approx 3.36
\]

Would you like further details or clarifications on any of these steps?

---

Here are 5 follow-up questions for you to consider:

1. Can you convert the final fraction \( \frac{2419}{720} \) into a mixed number?
2. How would this problem change if you replaced division with multiplication?
3. What is the decimal approximation of \( \frac{29}{6} \div \frac{24}{7} \) alone?
4. What if the numbers were in scientific notation—would it be easier to solve?
5. Can you think of a real-world scenario where this calculation might be useful?

**Tip:** Always convert mixed numbers into improper fractions when dealing with division or multiplication problems to simplify the process.

Learn how to solve the arithmetic expression 3.9 x 0.5 + 4 5/6 ÷ 3 3/7 with a detailed step-by-step solution. This problem covers key arithmetic concepts, including multiplication of decimals, division of fractions, and addition of fractions. Suitable for students in Grades 6-8.

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